Journal Club Theme of October 2009: Peridynamics applied to the structure and evolution of discontinuities

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Discontinuities have a fundamental role in the mechanics of solids. The most famous type of discontinuity is a crack, but others are important too, such as dislocations and phase boundaries. Many types of deformation that appear to be continuous at the macroscale, such as plastic flow in metals, really involve the evolution of discontinuities at some smaller scale.

How should we model discontinuities within a continuum framework? This is a long-standing question. Let’s first focus on cracks. The PDEs of the standard theory of solid mechanics cannot be applied directly at the points in a continuous body along which a crack grows, because the necessary spatial derivatives of the deformation do not exist there. (Using the weak form of the PDEs does not resolve this issue.)Therefore, the standard approach to modeling cracks is to introduce extra equations into the standard theory. A separate kinetic relation, extraneous to the basic field equations, is supplied to determine how a crack grows. This approach brings with it the following challenges and questions, among others:

What determines whether a crack should nucleate, i.e., what mathematical conditions lead to the spontaneous appearance of a discontinuity within a previously continuous body?

How can we determine, in sufficient generality, the correct kinetic relation that governs crack motion? How fast does a crack grow? In what direction should it grow? Should it branch? Should it change mode? Should it oscillate? Should it arrest? What should all these things depend on? How do we account for interfaces and defects close enough to the crack tip to affect the asymptotic fields? The science of fracture mechanics is largely concerned with these questions, but the answers it provides are sometimes limited to ideal conditions.

In a computational method designed to solve the standard PDEs in a continuous body, how do we keep track of all the new surfaces created by crack growth? How do we ensure that the discretization does not artificially introduce preferred directions for crack growth? How do we know that the method converges to some exact solution?

The peridynamic formulation of solid mechanics attempts to address these issues by replacing the standard PDEs with new field equations that, we hope, are better suited to the study of discontinuities. These field equations, which are integro-differential equations, can be applied directly on discontinuities. Cracks nucleate, initiate, and grow spontaneously according to the equation of motion and constitutive model, which do not involve the spatial derivatives of the deformation. Cracking is treated as just another form of deformation rather than as a separate phenomenon that requires separate equations. Cracks “do whatever they want.”

I would like to refer to four papers that demonstrate the potential of the theory and some of its current limitations. The first of these papers demonstrates unguided crack growth in a peridynamic solid. In the problems studied in this paper, the driving force for crack growth comes from a temperature gradient.

This next paper applies the peridynamic method to composite laminates. It demonstrates the influence of fiber directions on the direction of crack growth in the presence of strongly anisotropic properties:

The following paper applies the method not to cracks, but to martensitic phase boundaries. It demonstrates that a kinetic relation for the motion of a phase boundary is implied by the peridynamic field equations. The method also predicts certain structural features within a phase boundary, as well as complexity in the evolution of phase boundaries in multiple dimensions.

K. Dayal and K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics, Journal of the Mechanics and Physics of Solids 54 (2006) 1811–1842 (http://dx.doi.org/10.1016/j.jmps.2006.04.001).

The capabilities of the peridynamic method, particularly the generality of material response that can be modeled within it, are evolving rapidly. Although the peridynamic model has pros and cons that can be debated, I hope that the community will consider the basic question that it tries to address to be worthy of discussion: Is the standard PDE-based continuum theory the best possible tool for modern solid mechanics?

In addition to the Dr. Silling's post, I just want to add
more information about the Peridynamic Theory. As Dr.Silling mentioned, the
classical continuum mechanics confronts problems due to its mathematical
structure if there is a discontinuity in the body. So, some people tried to
overcome this problem within the classical approach and some others proposed
non-classical approaches such as non-local theory of elasticity. Peridynamic
theory is a sub-class of the non-local theory. So, in order to understand the Peridynamic
theory better, it is a good idea to look at the difference between the
classical continuum theory and non-local theory of elasticity.

As we all know that in continuum mechanics we are making an
assumption that a body is composed of infinitesimally small material volumes
that we call material points. According to the classical continuum mechanics,
there is an assumption about how these material points are interacting with
each other. Cauchy proposed that a material point can only interact with the
material points in its nearest neighborhood and these interactions were defined
as "Traction vectors". After relating the traction vector to the stress tensor
and by utilizing the conservation of linear momentum, one can obtain the
well-known governing equation of the classical continuum theory for a material
point i,

sij,j
+ bi = rui,tt (1)

As you can see Eq. (1) is a partial differential equation
and the derivatives are not defined if there is a discontinuity in the
structure.

As opposed to the classical continuum mechanics, in non-local
theory, the interactions between material points are not limited to the nearest
neighbors, so all material points can interact with all other material points
in the body. In peridynamic theory, these interactions are defined as forces
that material points are exerting on each other by following a very similar
idea used in molecular dynamics, i.e. interaction of atoms with each other. But,
please note that in peridynamic theory material points are interacting with
each other, not atoms. By using this idea, Dr. Silling proposed the governing
equation of the peridynamic theory for the material point i

Integral(f dV)+ bi = rui,tt (2)

where f corresponds to the force (per unit volume squared)
that all material points are exerting on the material point i. As you can see,
Eq. (2) is an integral equation rather than a differential equation. So, this
equation is valid whether there is a discontinuity in the structure or not.

For those of you who are new to the idea of Periydnamic
Theory, you can start by reading this paper:

The update on peridynamics was quite interesting. However, I still don't have a clear idea on what's holding the method back from widespread acceptance. Could you give us some pointers? I haven't yet been able to get a peridynamic theory based program funded.

I'd also like to point out some earlier discussions of the method on iMechanica.

Let me try to answer your question on “what's holding the method back from widespread acceptance”.

As we all know classical continuum mechanics is a well-established
method that we can utilize to obtain the deformation response of objects due to
applied forces. Tremendous number of problems has been solved by using
classical continuum mechanics with a very high satisfaction in accuracy of
results. Although the method is successful in many different type of problems,
if there is some type of discontinuity in the structure, the method started to
fail. As a result of this, the field of “fracture mechanics” was established. Today,
in order to predict the failure in structures, different techniques are used
such as XFEM, cohesive zone models etc. And the techniques that I just
mentioned are based on classical continuum mechanics. It looks like to me,
today, XFEM is the most popular method of these techniques and probably the
main rival of the peridynamic theory in predicting failure in structures. I
think we need to ask a question at this point: “Why XFEM is more popular than
Peridynamic Theory?”. I think the answer is simple. First of all XFEM is a
finite element technique and almost every solid mechanician has some idea about
what finite element method is all about. In other words, people have some
familiarity with the topic. Even today, Abaqus included this approach as part
of the software. On the other hand, peridynamics is a pretty new method and
first peridynamic paper appeared in 2000. Its structure is also different than
most solid mechanicians are used to. It has a very similar structure to
molecular dynamics and I don’t think many of the solid mechanicians are
familiar with it. But, I am optimistic about the widespread acceptance of
Peridynamic theory in the future, because today around 20 journal papers based
on peridynamics were published and by the time the people start to understand
its logic, they’ll start to accept it and utilize it maybe not only for
macro-scale problems but also at nano-scale.

Regarding to the funding, as far as I know, there are some
peridynamics based projects already funded by different sources.

It is said that a meshfree code by peridynamci method is avilable now. When I searched to http://www.sandia.gov/emu/emu.htm, I can not find the download. Would you please kindly tell me where to get it?